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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On periodicity of entire functions


Author: Chung Chun Yang
Journal: Proc. Amer. Math. Soc. 43 (1974), 353-356
MSC: Primary 30A64
MathSciNet review: 0333180
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Abstract: A sequence $ S = \{ {s_n}\} $ is said to be a periodic set of period $ \tau ( \ne 0)$ if and only if $ {S^\ast } = \{ {s_n} + \tau \} $ can be rearranged to be a sequence to coincide with $ S$. Let $ F$ be the class of all entire functions $ f$ satisfying the growth condition:

$\displaystyle \mathop {\lim }\limits_{r \to \infty } \sup \log \log \log M(r,f)/\log r < 1.$

In this paper it is shown that if $ f \in F$ and the zero sets of $ f$ and $ f'$ both are periodic sets with the same period $ \tau $, then $ f$ can be expressed as $ f(z) = {e^{cz}}g(z)$, where $ c$ is a constant and $ g(z)$ is a periodic entire function with period $ \tau $. A counterexample is exhibited to show that the above condition is a necessary one.

References [Enhancements On Off] (What's this?)

  • [1] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 0164038
  • [2] Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330

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DOI: https://doi.org/10.1090/S0002-9939-1974-0333180-1
Article copyright: © Copyright 1974 American Mathematical Society